Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| socle | C2 | C22×AΣL(1,𝔽16) |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,15,30)(4,10)(5,19)(6,12,22,28)(7,9)(8,14,25,31)(11,21,27,18)(16,24,17,29)(20,26,32,23);(1,3)(2,8)(4,14)(5,20)(6,15)(7,17)(9,23)(10,24)(11,29)(12,25)(13,27)(16,22)(18,31)(19,30)(21,32)(26,28);(2,9)(8,23)(10,19)(11,27)(12,28)(13,29)(24,30)(25,26);(1,4,7,5)(2,10,9,19)(3,14,17,20)(6,18,22,21)(8,24,23,30)(11,28,27,12)(13,25,29,26)(15,31,16,32)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fg,g^b=gi,h^b=hi,f^e=fhi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279319, #72279376, #72279397, … |
| maximal normal subgroup of | #72279319, #72279376, #72279397, … |
| maximal quotient of | #72306380, #72340590, #72340596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419774);